Proof about operations on sets

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SUMMARY

The discussion focuses on proving that the union of two indexed families of sets, specifically (∪r ∈ ℝ Ar) ∪ (∪r ∈ ℝ Br), is a subset of the union of the sets Ar and Br for all r in the real numbers. Participants emphasize the need to demonstrate that if an element x belongs to the left-hand side, it must also belong to the right-hand side, establishing the necessary subset relationship. The proof hinges on the definitions of unions and subsets within set theory.

PREREQUISITES
  • Understanding of set theory, particularly indexed families of sets.
  • Familiarity with the concepts of unions and subsets.
  • Basic knowledge of mathematical proofs and logical reasoning.
  • Proficiency in notation involving real numbers and set operations.
NEXT STEPS
  • Study the properties of indexed families of sets in set theory.
  • Learn about the formal definitions of unions and subsets in mathematical logic.
  • Explore techniques for constructing mathematical proofs, particularly direct proofs.
  • Review examples of set operations involving real numbers to solidify understanding.
USEFUL FOR

Mathematics students, educators, and anyone interested in advanced set theory and mathematical proofs will benefit from this discussion.

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Homework Statement


Let the set Ar: r is an element of all real numbers and the set Br: r is an element of all real numbers be two indexed families of sets.

Prove that (upside U r is an element of the reals Ar) U (upside U r is an element of the reals Br) is a subset of upside U r is an element of the reals (ArUBr).


Homework Equations





The Attempt at a Solution



Could someone please give me a hint? I know that Ar and Br have to be unions of each other and at the same they are subsets of the other one, but I don't really know how I would go about proving this.

Thank you much
 
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That's kind of hard to read. But if x is an element of the left hand side, then it's either in all of the A_r or it's in all of the B_r. If x is in the right hand side then it's in either A_r or B_r for all r. Assume each of the conditions on the left and prove they imply the right.
 
Thank you very much

Regards
 

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